StressStrain Theory. Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory. . u(x+dx). dx. dx’. dx. dx’. u(x). x. x’.
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StressStrain Theory
Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory.
u(x+dx)
dx
dx’
dx
dx’
u(x)
x
x’
Length squared: dl = dx + dx + dx = dx dx
2
2
2
2
i
i
1
2
3
dl = dx’ dx’ = (du +dx )
2
2
i
i
i
i
= du du + dx dx + 2 dudx
i
i
i
i
i
i
Strain Tensor
After deformation
Displacement vector: u(x) = x’ x
u(x+dx)
dx
dx’
dx
dx’
u(x)
x
x’
Length squared: dl = dx + dx + dx = dx dx
2
2
2
2
i
i
1
2
3
dl = dx’ dx’ = (du +dx )
2
2
i
i
i
i
Length change:dl  dl = du du + 2du dx
2
2
= du du + dx dx + 2dudx
i
i
i
i
i
i
i
i
i
i
du = du dx
Substitute
i
i
j
dx
j
Strain Tensor
After deformation
(1)
into equation (1)
u(x+dx)
dx
dx’
dx
dx’
u(x)
x
x’
(1)
into equation (1)
Length change:dl  dl = U U
2
2
Strain Tensor
Length change:dl  dl = du du + 2du dx
i
i
2
2
i
i
i
i
(du + du + du du )dx dx
du = du dx
=
i
j
j
k
k
i
Substitute
dx
dx
dx
dx
i
i
j
dx
j
j
i
i
j
Strain Tensor
After deformation
(2)
1 light year
Problem
V > C
1 light year
Problem
V > C
V < C
Elastic Strain Theory
Elastodynamics
dL L’L
e
=
=
=
Length Change
L L
xx
Length
L’
L’
L
L
Acoustics
dL L’L
e
=
=
=
Length Change
L L
xx
Length
L’
L’
L
L
Acoustics
dL L’L
e
=
=
=
Length Change
L L
xx
Length
L’
L’
L
L
Acoustics
No Shear Resistance = No Shear Strength
Tensional
dw
du
dz
Acoustics
dw, du << dx, dz
dx
dxdz+dxdw+dzdudxdz
AreaChange
(dz+dw)(dx+du)dxdz
=
=
+ O(dudw)
dx dz
dx dz
Area
dw du
=
+
dz
dx
e
+
dw
=
du
xx
U
=
dz
e
zz
Acoustics
really small
big +small
big +small
Infinitrsimal strain
assumption: e<.00001
Dilitation
dx
k
e

P =
(
)
+
Bulk Modulus
xx
U
=
e
dx
zz
1D Hooke’s Law
pressure
strain
k
du
Infinitrsimal strain
assumption: e<.00001
F/A =
Pressure is F/A of outside
media acting on face of box
k
F/A =
(
)
+
xx
Compressional
Source or Sink
k
Larger = Stiffer Rock
=
e
e
zz
zz
Hooke’s Law
Dilation
e
k
U
Infinitrsimal strain
assumption: e<.00001
e
k

P =
(
)
+ S
+
xx
Bulk Modulus
Newton’s Law
..
..


dP
dP
r
r
;
w =
u =
dx
dz
Net force = [P(x,+dx,z,t)P(x,z,t)]dz
density
x,
..
k
Larger = Stiffer Rock
r
u
P (x,z,t)
P (x+dx,z,t)
ma = F
dxdz
Newton’s Law
1stOrder Acoustic Wave Equation
..
P

r
u =
u=(u,v,w)
..
..


dP
dP
r
r
;
w =
u =
dx
dz
density
k
Larger = Stiffer Rock
P (x,z,t)
P (x+dx,z,t)
Newton’s Law
1stOrder Acoustic Wave Equation
..
P

r
u =
(1)
(3)
(4)
..
..
k
P
= 
U
(2)
..
]
P

[
u =
1
r
..
k
]
P

[
P =
1
r
(Newton’s Law)
(Hooke’s Law)
Divide (1) by density and take Divergence:
Take double time deriv. of (2) & substitute (2) into (3)
Newton’s Law
2ndOrder Acoustic Wave Equation
..
k
]
P

[
P =
1
r
..
k
P
P =
r
k
c =
2
Substitute velocity
r
..
2
P
c
P =
2
Constant density assumption
Summary
..
..
P

k
r
k
]
u =
P

[
1. Hooke’s Law: P
P =
= 
U
1
r
2. Newton’s Law:
3. Acoustic Wave Eqn:
k
c =
2
r
..
2
P
c
;
P =
2
Constant density assumption
Body Force Term
+ F
Problems
1. Utah and California movingEW apart at 1cm/year.
Calculate strain rate, where distance is 3000 km. Is it e or e ?
2. LA. coast andSacremento moving NS apart at 10cm/year.
Calculate strain rate, where distance is 2000 km. Is is e or e ?
xx
xx
xy
xy
3. A plane wave soln to W.E. is u= cos
(kxwt) i.
Compute divergence. Does the volume change
as a function of time? Draw state of deformation boxes
Along path
U
U
n
dl
U
= lim
k
e

A
P =
(
)
+
A 0
xx
+ U(x,z+dz)cos(90)dx
+ U(x,z+dz)cos(90)dx
 U(x,z)dz
dxdz
dxdz
dxdz
dxdz
(x+dx,z+dz)
n
n
e
zz
Divergence
= U(x+dx,z)dz
>> 0
= 0
No sources/sinks inside box.
What goes in must come out
Sources/sinks inside box.
What goes in might not come out
U(x,z)
U(x+dx,z)
(x,z)